Binomial expansion (negative index)

Series

A-level Maths (9MA0)

ORIGINAL

The general binomial series — mind the signs and remember to raise the whole x-term to each power.

Verified against Edexcel 9MA0 (2026 spec)

Question

Find the first three terms, in ascending powers of $x$ , of the binomial expansion of $(1 + 3x)^{-2}$ .

Every step worked, with the reasoning.

1
$(1 + X)^n = 1 + nX + \dfrac{n(n-1)}{2!}X^2 + \dots$ with $n = -2$ , $X = 3x$ .
Use the general binomial series and identify $n$ and the $x$ -term.
2
$= 1 + (-2)(3x) + \dfrac{(-2)(-3)}{2!}(3x)^2 + \dots$
Substitute $n = -2$ and $X = 3x$ into the formula.
3
$= 1 - 6x + \dfrac{6}{2}(9x^2) + \dots$
Evaluate: $(-2)(-3) = 6$ and $(3x)^2 = 9x^2$ .
4
$= 1 - 6x + 27x^2 + \dots$
Simplify the $x^2$ coefficient: $3 \times 9 = 27$ .

Answer: $1 - 6x + 27x^2$

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